Fingerprints of Water
An interactive primer on δ-notation, isotope fractionation, the meteoric water lines, deuterium excess, and the decay law — the physics that writes the signals every tracer study reads. Builds on Week 1; no isotope background needed.
Rain on Barbados and snow at 82°N are both H₂O. Can anyone tell them apart?
Strip away the salts, the dust, and the dissolved gases. Pure water from the tropics and pure water from the Greenland coast — is there anything left to measure?
Ordinary water is mostly ¹H₂¹⁶O, but about 2,000 molecules in every million carry a heavy oxygen (¹H₂¹⁸O) and about 320 in a million carry a heavy hydrogen (¹HD¹⁶O) — the proportions Dansgaard (1964) quoted in the first pages of the paper you are reading this week. Those proportions are not fixed. Tropical island rain holds about 0.2% fewer heavy-oxygen molecules than the ocean; the monthly snowfall at Station Nord (82°N) holds up to about 3% fewer. In the language you will learn in Section 3, that is δ¹⁸O ≈ −1.6‰ for the islands and −9 to −33‰ for Nord (Dansgaard, 1964, Tables 10–11).
Week 1 defined a tracer as a signal plus a transport story plus a model of interpretation. This week supplies the physics behind the signal — and the first two ways a tracer can stop being conservative: reaction and radioactive decay.
Water comes in light and heavy versions
Hydrogen has two stable isotopes (¹H and ²H, "deuterium", D) and one radioactive one (³H, tritium). Oxygen has three stable isotopes (¹⁶O, ¹⁷O, ¹⁸O). Combine them and you get a family of water molecules — isotopologues — that differ only in mass. Click a molecule.
Pick a molecule above to see why it matters.
Stable isotopes: a signal, not a clock
- They never disappear. ²H and ¹⁸O are stable; their proportions change only when physics sorts molecules by mass — most importantly when water changes phase.
- They ride inside the water molecule. Nothing has to dissolve; the label is the water itself, so below ground it travels with the water (Kendall et al., 2014).
- They are cheap to measure. A modern laser analyser or mass spectrometer resolves ~0.1–0.2‰ in δ¹⁸O and ~1‰ in δ²H.
Tritium: the same molecule, with a clock
- ³H decays (half-life 12.32 years), so its abundance falls predictably with time — the basis of the bomb-pulse dating you met in Week 1.
- It is vanishingly rare: 1 tritium unit (TU) is one ³H per 10¹⁸ hydrogen atoms; its mass effect on fractionation is irrelevant at that abundance.
- Section 10 returns to the decay law that governs it.
δ: a ratio of ratios, reported in parts per thousand
Nobody can measure the absolute number of ¹⁸O atoms in a sample precisely, but a mass spectrometer measures the difference between a sample and a reference water superbly. So the convention is to report that difference, relative to the reference, in per mil (‰):
where R is the ratio of heavy to light isotope (¹⁸O/¹⁶O or ²H/¹H). The reference is SMOW — "Standard Mean Ocean Water", defined by Craig (1961b) from a reference water held by the U.S. National Bureau of Standards and the ocean measurements of Epstein & Mayeda (1953). Today the scale is anchored by its successor, VSMOW (²H/¹H = 155.76 × 10⁻⁶; ¹⁸O/¹⁶O = 2005.2 × 10⁻⁶), with a second water, SLAP, fixing the far end of the scale at δ²H = −428‰ and δ¹⁸O = −55.5‰.
Count the heavy atoms
Imagine counting one million oxygen atoms in the reference water: about 2,005 of them are ¹⁸O. Now count a million in your sample. The slider sets how many you find; the readout converts the count to δ. Switch to hydrogen to do the same with deuterium.
Sign language
δ < 0: the sample has fewer heavy atoms than the ocean ("depleted", "lighter"). δ > 0: more ("enriched", "heavier"). Almost all fresh water is negative, because the ocean is where the water cycle starts and each evaporation–condensation step strips heavy isotopes preferentially.
Precision and range
Dansgaard's laboratory measured to ±2‰ in δ²H and ±0.2‰ in δ¹⁸O, against a natural range of more than 400‰ and 40‰ — "at least 200 times the measuring accuracy" (1964, §1.2). Today's laser analysers do about as well, and the Picarro session in Week 8 will show you where that precision comes from and where it fails.
A small trap
δ values are not simply additive. If a sample is δ′ relative to a working standard that is itself δst relative to SMOW, then δ = δ′ + δst + δ′δst/1000 (Dansgaard, 1964, eq. 1). The cross term is tiny for small values and matters for large ones — one reason laboratories calibrate on the VSMOW–SLAP scale rather than adding offsets.
Fractionation: heavy molecules are slower to leave
When liquid water and its vapour sit in equilibrium, the vapour is poorer in heavy isotopes than the liquid, because the heavy molecules have the lower vapour pressure. The size of that difference is the fractionation factor:
At "normal temperature" Dansgaard quotes α ≈ 1.08 for HDO and ≈ 1.009 for H₂¹⁸O: vapour in equilibrium with the ocean is depleted by some 80‰ in deuterium and 9‰ in ¹⁸O. The factor depends on temperature — colder means larger — which is what turns isotopes into thermometers. Drag the slider.
Equilibrium fractionation versus temperature
Values interpolated from Dansgaard's Table 1 (Merlivat et al., 1963 for deuterium; Zhavoronkov et al., 1955 for ¹⁸O; extrapolated below 0 °C). Modern determinations are slightly different — about 1.084 and 1.0098 at 20 °C (Horita & Wesolowski, 1994) — but the pattern is identical.
And the first drop of rain? Condense a little of that vapour back to liquid at the same temperature and it is enriched by the same factor — so the first condensate from ocean-equilibrium vapour has exactly the ocean's composition, δ = 0 (Dansgaard's eq. 5). Everything lighter than the ocean comes from what happens next: Section 5.
Equilibrium fractionation
- Slow and reversible. Molecules cross the interface in both directions; the ratio settles to a value fixed by temperature alone.
- Where it holds: condensation inside clouds, evaporation into saturated air, a raindrop falling through air at 100% humidity.
- Signature: deuterium and ¹⁸O move together in a ratio near 8 — the origin of the meteoric water line's slope.
Kinetic fractionation
- Fast and one-way. When vapour is removed faster than it can exchange — evaporation into dry, windy air, diffusion through still air — the lightest molecule escapes fastest: c(H₂¹⁶O) > c(HDO) > c(H₂¹⁸O) (Dansgaard's eq. 14).
- It hits ¹⁸O harder than deuterium. In the thin-water-jet evaporation experiments Dansgaard reports, the ¹⁸O effect grew by ~200% while the deuterium effect changed by no more than 15%. The δ²H/δ¹⁸O slope collapses from ~8 to ~3–5.
- Signature: waters that have evaporated fall below the meteoric water line. Ocean vapour formed this way carries a surplus of deuterium — the deuterium excess of Section 7.
Rainout: a cooling air mass distils itself
Precipitation forms when moist air cools. The first condensate takes more than its share of heavy isotopes, leaving the remaining vapour lighter; the next condensate forms from that lighter vapour, and so on. If each increment of condensate is removed as it forms — the Rayleigh condition — the isotopic composition runs away toward very negative values as the vapour is used up:
where Fv is the fraction of the initial vapour still remaining, α₀ is the fractionation factor at the starting temperature t₀, α at the current temperature, and αm at the mean of the two. Dansgaard computed Fv from the saturation mixing ratio of the air as it cools at constant pressure — the same is done here.
Cool an air mass and watch its rain get lighter
Vapour leaves the ocean in equilibrium at the dew point t₀ (its first condensate is SMOW, 0‰). Drag the temperature down and read the composition of the newly forming condensate and the remaining vapour. Presets reproduce Dansgaard's C₀, C₂₀ and C₃₀ curves (Fig. 5).
Composition along the cooling path
condensate (blue) and remaining vapour (aqua) as the air mass cools from t₀
The same path in δ²H–δ¹⁸O space
the C-curve of new condensate; grey line = δ²H = 8 δ¹⁸O + 10
Check against Dansgaard's Table 2
Isobaric cooling from t₀ = 20 °C. Dansgaard: at 0 °C, Fv = 26%, δ²H = −95‰, δ¹⁸O = −11.7‰; at −20 °C, Fv = 5.2%, δ²H = −223‰, δ¹⁸O = −28.2‰. The table below is this page's calculation with the same fractionation factors.
Why the slope is ~8
Both isotopes obey the same equation; only α differs. Over 40° of cooling from 20 °C the δ²H–δ¹⁸O track is nearly straight with mean slope 8.0 ± 0.2 (Dansgaard's Table 5) — not the naive ratio (αD − 1)/(α18 − 1), which would give 10.9, because α itself changes along the path (eq. 8). Craig's global slope of 8 is Rayleigh condensation made visible.
Two extremes, one reality
Rayleigh removal is one limit; condensate that stays and re-equilibrates with the vapour (a closed system) is the other and fractionates far less (Dansgaard's Fig. 1; Kendall et al., 2014, Fig. 1, for the analogous evaporation case). "In nature, exchange will, more or less, smooth out the phenomenon" — real clouds sit between the curves.
Temperature, not thermometer
A single rain cannot be read as a condensation temperature — its δ depends on the starting vapour, the dew point, the degree of cooling and the way of cooling (Dansgaard, §2.5). Only long-term means smooth the scatter enough to correlate with temperature. That is Section 6.
Temperature, amount, latitude, altitude, continent: Dansgaard's "effects"
The IAEA–WMO precipitation survey (begun 1961; over one hundred stations by 1964) let Dansgaard compare monthly rain across the world. Out of the scatter came a handful of regularities that every hydrologist still uses. The names are his.
Temperature effect
Annual mean δ¹⁸O of precipitation falls linearly with mean annual air temperature — about 0.7‰ per °C across North Atlantic coastal stations and the Greenland ice cap (eq. 9). It also drives the seasonal cycle at high-latitude continental stations: summer rain is 5–10‰ heavier than winter snow at Chicago, Edmonton, Nord (Table 12).
Amount effect
In the tropics, where temperature barely varies, rainy months are isotopically light and sparse months heavy: about −1.6‰ per 100 mm at tropical islands and −2.0‰ at partly continental stations (§4.2.3). Deep cooling in big storms, plus less evaporation and exchange from falling drops in humid air, explain it.
Latitude, altitude, continent
Island groups average −1.6‰ at 0–23°, −3.4‰ at 23–45°, −6.6‰ at 45–90° (Table 11). Rain gets lighter going inland (Valentia → Stuttgart → Vienna) and over mountains (Holsteinsborg → Søndre Strømfjord). Elevation gradients are typically −0.1 to −0.5‰ per 100 m in δ¹⁸O (Jasechko, 2019).
Station explorer — Dansgaard's Table 10, one station at a time
Each station's monthly samples (1961–62) define a δ¹⁸O range, a δ²H–δ¹⁸O slope, a deuterium surplus d, and whether a temperature or amount effect was visible. Pick a station; the small plot shows its line against Dansgaard's reference line L (δ²H = 8 δ¹⁸O + 10), and the note explains what Dansgaard read into it.
The temperature effect, as Dansgaard drew it (Fig. 3)
Slide the mean annual air temperature; the line is his regression through 38 stations — 17 North Atlantic continental stations, 6 islands, and 15 Greenland and Antarctic ice-cap sites — spanning some 75 °C. The curve "should not exceed δ¹⁸O = 0 (SMOW), since positive δ's occur only as a result of special processes like accidental evaporation from liquid precipitation."
Points: island-group means from Table 11; Station Nord, Horlick Mountains and the South Pole as plotted in Fig. 3.
Observed slope ≈ 0.70‰/°C; a simple Rayleigh process starting at 20 °C predicts 0.66–0.67‰/°C (eq. 8). "As a simplified model the Rayleigh condensation is, apparently, sufficient for the interpretation of most observations" — but note the continental effect: for ta above about −5 °C, inland stations fall below the line.
The δ²H–δ¹⁸O plane and the meteoric water lines
Plot δ²H against δ¹⁸O and the world's precipitation collapses onto a line. Craig (1961) drew it from about 400 samples of rivers, lakes, rain and snow from many parts of the world: δ²H = 8 δ¹⁸O + 10, for waters "which have not undergone excessive evaporation". Dansgaard's Northern Hemisphere continental stations gave 8.04 δ¹⁸O + 9.5 (unweighted) and 8.1 δ¹⁸O + 11 (weighted means); a modern compilation of 68,382 precipitation samples gives 7.93 δ¹⁸O + 8.99 (Jasechko, 2019). The slope is Rayleigh condensation; the intercept is something else.
Walk a parcel of water around the diagram
Follow the steps. Each one applies a process from Sections 4–5 to the water and moves it in the diagram; the deuterium excess d tells you which kind of process it was. Hover anywhere on the plot to read δ and d; click to drop a sample marker.
What d-excess records
- Equilibrium processes do not change d. Condense, re-condense, cool further: the point slides along a slope-8 line and keeps its d (Dansgaard, §3.1).
- Kinetic evaporation from the ocean raises the vapour's d, the more so the drier and windier the air. Dansgaard called it a "rate of evaporation index" of the source region: Northern Hemisphere continental rain averages d ≈ +9.4‰; Africa and the Near East +14.8, South America +12.8, Australia +14.1 — vapour formed "farther from equilibrium" (Table 9).
- Kinetic evaporation from a limited water body lowers d of what remains: lakes, soil water, falling drops in dry air. Only one of more than a hundred stations had negative d: Horlick Mountains, Antarctica (−2.8‰).
Slopes you will meet
- ≈ 8: equilibrium condensation (and equilibrium evaporation: Dansgaard's sE ≈ 8–9). Most of Dansgaard's stations: 8 ± 1.
- ≈ 4–5 through SMOW: first-stage rain at tropical islands, δ²H = 4.6 δ¹⁸O + 0.1 (eq. 15) — condensate from vapour with varying kinetic surplus.
- ≈ 2–6, off the line: evaporation from open water or soil (Craig's closed basins ≈ 5; Lake Victoria and the Nile ≈ 5; Jasechko's compilation ~3–6, soils ~2–5, humid open water ~5–8).
- > 8: rare — Entebbe and Addis Ababa, where light rains exchange with rapidly evaporated fresh water (Fig. 20).
Below the line: a global test of "two water worlds"
Evaristo, Jasechko & McDonnell (2015) used nothing more than Sections 4–7. Precipitation, stream water and groundwater at a site plot along the local meteoric water line; water that has evaporated plots below it. So if the water that plants transpire has a different evaporation history from the water that becomes groundwater and streamflow, the two should separate in the diagram. They compiled dual-isotope data from 47 sites in five biomes — plant xylem water (n = 1,460), soil water (1,830), stream water (336), groundwater (2,749) and precipitation (488) — and measured each water's distance from its LMWL:
with a and b the slope and intercept of the site's LMWL and S the combined analytical uncertainty (≈1‰). Local precipitation has an offset of zero by definition; evaporation drives it negative.
The second tool traces an evaporated water back to the precipitation it came from: extend its evaporation line (slope m) until it meets the LMWL. The meeting point solves the two line equations simultaneously:
Equations (2)–(3) as printed in the 2015 Letter do not perform this intersection: they project the evaporation line's axis intercept onto the LMWL, which lands off the evaporation line and biases the inferred source heavy. The demonstration below computes the intersection directly; note that it lies down-line of the sample only when m < a.
Build a site and read the offsets
An idealized site: groundwater and stream water sit on the LMWL; soil water has evaporated along a shallow line; plant xylem water lies on the soil's evaporation line, bounded by soil water — as the compilation found at site level (Extended Data Fig. 4). Adjust how much the soil evaporated, how shallow its evaporation slope is, and whether groundwater recharged from different rain than the plants used.
Separation
At 40 of 47 sites, groundwater offsets differed statistically from both soil and xylem offsets; groundwater and stream water sat on average 5.4 and 4.8 offset units closer to the LMWL. The gap was largest in the tropics (7.7) and Mediterranean (5.4), smallest in temperate forests (1.6) and grasslands (2.4).
Segregation
Tracing xylem water back along its evaporation line to the LMWL gives the precipitation it came from. At 80% of sites (37 of 46) that source differed from groundwater (median δ²H −82‰ vs −52‰): plants and aquifers were fed by different rain, "segregated in space and time", before the subsurface separated the waters further.
Why it matters
Land-surface models and isotope-based transpiration/evapotranspiration estimates assume one well-mixed soil reservoir feeds plants, recharge and streams. If the compartments differ, downstream isotopes are biased toward precipitation and groundwater and "do not reflect the composition of water seen in soil" — Weeks 3, 11 and 13 return to the consequences.
Conservative, reactive, or decaying?
A conservative tracer changes only by mixing: it moves with the water and nothing produces or consumes it on the way. A reactive tracer is transformed en route — a nuisance for source questions, a gift for process questions. A decaying tracer is lost at a rate fixed by physics, which is a predictable kind of non-conservation. Every tracer carries an assumption; classify each one and see the assumption it carries.
The decay law: a clock that ignores temperature, pressure and chemistry
Radioactive decay is probabilistic: in any small interval dt each unstable nucleus has the same chance λ dt of decaying, regardless of its history or surroundings. Among N nuclei the loss rate is therefore proportional to N — a first-order rate law from which "essentially all the significant equations of radiogenic isotope geochemistry and geochronology can be derived" (White, 2015, eq. 1.12):
Pick a clock and run it
Each radionuclide used this term has its own half-life and therefore its own useful window — very roughly a tenth of a half-life to five half-lives, depending on how precisely the remaining fraction can be measured and how well the starting amount N₀ is known. Choose a nuclide and drag the time slider.
What makes decay a clock
- λ is a constant of the nuclide. Temperature, pressure and chemistry do not touch it (electron capture shows a negligible pressure dependence — White, 2015). Fractionation, by contrast, depends on temperature and on how fast a process runs.
- The remaining fraction is all you measure. Turning it into a date needs N₀ — the input — and a transport model: the bomb-pulse lesson of Week 1, made quantitative in Weeks 3 and 5.
- Mixtures break the arithmetic. The mean of two ages is not the age of the mixture's mean activity. This is why "apparent age" is a distinct concept (Week 3).
Stable and radioactive, side by side
- δ¹⁸O, δ²H: stable; the signal is set at recharge and changes only by mixing (or phase change). No clock — but an origin, a pathway, a process.
- ³H: the same water molecule, with a 12.32-year clock; tritium-helium (Week 5) recovers N₀ by counting the daughter.
- ¹⁴C, ³⁶Cl, ³⁹Ar, ⁸¹Kr, ³⁵S, ⁸⁵Kr: clocks of different lengths, each carried by a different solute or gas with its own reactivity — why Section 9's classification and Section 10's clock have to be read together.
Three papers, three jobs
Dansgaard (1964) built the physics; Craig (1961) drew the reference line; Evaristo et al. (2015) used both to test a hypothesis about where plants get their water. The guides below are maps, not substitutes: each says what the paper asks, where to find the load-bearing figures, which equations to recognize, and what a careful reader might press on. The notes and the critique are yours to write.
Dansgaard, W. (1964). Stable isotopes in precipitation. Tellus, 16(4), 436–468.
Thirty-three pages that go from the vapour pressure of HDO to the climate of Tokyo. Read chapters 1–2 slowly (the physics), skim the station-by-station detail of chapter 4 for the figures named below, and do not skip chapter 5 — it is two pages on hydrology. doi:10.1111/j.2153-3490.1964.tb00181.x (open access)
Craig, H. (1961). Isotopic variations in meteoric waters. Science, 133(3465), 1702–1703.
Two pages and one figure, cited thousands of times. Read it with its companion note, Craig (1961b), which defined the SMOW standard the δ values are reported against. doi:10.1126/science.133.3465.1702
Evaristo, J., Jasechko, S., & McDonnell, J. J. (2015). Global separation of plant transpiration from groundwater and streamflow. Nature, 525, 91–94.
A four-page Letter with a long Methods section and six Extended Data figures and three tables; the argument is in Figures 1–2 and equations 1–3, and the vulnerabilities are discussed in Methods. doi:10.1038/nature14983
Read four samples
Each scenario gives you the kind of numbers a Week 2 reader should now be able to interpret. Pick the reading you would defend first, then compare with the debrief.
Nine questions before Week 3
Immediate feedback, no grade, no record. If you miss one, the linked section is the fix.
Glossary
The Week 2 working vocabulary, plus a few terms you will meet in Week 3. Search or browse.
The Week 2 readings — and two optional companions
Papers are not posted for this course: retrieving them from the citation and DOI — via UGA Libraries, GALILEO, or the publisher — is part of the training.
Dansgaard, W. (1964). Stable isotopes in precipitation. Tellus, 16(4), 436–468.
Fractionation factors, Rayleigh processes, the temperature and amount effects, the δ²H–δ¹⁸O relation, kinetic effects and d, and the first reading of the IAEA–WMO survey. doi:10.1111/j.2153-3490.1964.tb00181.x
Craig, H. (1961). Isotopic variations in meteoric waters. Science, 133(3465), 1702–1703.
The global meteoric water line, δ²H = 8 δ¹⁸O + 10. Pair with Craig (1961b), Science 133(3467), 1833–1834, which defines SMOW. doi:10.1126/science.133.3465.1702
Evaristo, J., Jasechko, S., & McDonnell, J. J. (2015). Global separation of plant transpiration from groundwater and streamflow. Nature, 525, 91–94.
Stable isotopes as conservative tracers, and distance from the LMWL as a global diagnostic of ecohydrological separation. doi:10.1038/nature14983
White, W. M. (2015). Isotope Geochemistry, Ch. 1 — Atoms and nuclei: their physics and origins. · Putman, A. L., et al. (2019). A global perspective on local meteoric water lines. J. Hydrol.
White grounds the decay law and the nuclides (pairs with Study Guide 01); Putman shows how much local lines vary and why. Not discussed in a block.
Coming in Week 3 — Groundwater dating I: age concepts and radiometric methods
What "groundwater age" means; residence time as a distribution rather than a number; ¹⁴C, ³⁹Ar and ⁸¹Kr for old water. On the discussion table: Cook & Herczeg (2000, Ch. 1), Bethke & Johnson (2008), and the paired Evaristo et al. (2019, 2026) readings on age distributions in soil–plant systems. The decay law of Section 10 and the mixture lesson of Week 1 meet head-on.
Carry-forward question: for the tracer you are considering for your proposal, is it conservative, reactive, or decaying in your setting — and what observation would show you were wrong?